It follows from known results, namely, memoryless
determinacy of simple stochastic games ([1]}),
that for all our generalised arrival games
it suffices to consider deterministic “essentially memoryless” strategies
for a player \(i\)  given by
\(\mathit {Strat}_i:(V_i\times Q) \rightarrow V\) ,
which ignore the history in a partial play \(\pi \) , and only considers the current state
\((v,q)\)  in order to choose (deterministically) a move to the next vertex, \(v^{\prime }\) ,
such that \((v^{\prime },q) \in \mathit {Valid}(v,q)\) .
(Note that switching positions only change during transitions out of switching nodes.)
Indeed, we can view our instances
of generalised arrival as defining exponentially larger simple stochastic games over the state space \(V\times Q\) , because of the deterministic way the switching position \(q\)  updates with each transition.
